Problem: Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{t^2 + 12t + 20}{t^2 + 8t + 12}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{t^2 + 12t + 20}{t^2 + 8t + 12} = \dfrac{(t + 10)(t + 2)}{(t + 6)(t + 2)} $ Notice that the term $(t + 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t + 2)$ gives: $r = \dfrac{t + 10}{t + 6}$ Since we divided by $(t + 2)$, $t \neq -2$. $r = \dfrac{t + 10}{t + 6}; \space t \neq -2$